<h2>The ring of Siegel modular forms of degree 2 with respect to 
<a class="knowl-title" knowl="mf.siegel.group.gamma0">$\Gamma_0(2)$</a>
</h2>

<div class="literature">
  <ul>
    <li><span class="name">T.Ibukiyama:</span> On Siegel modular varieties of level 3. Internat. J. Math. 2 (1991), 17-35, <a href="http://www.ams.org/mathscinet-getitem?mr==0141643">MR1082834</a></li>
  </ul>
</div>

<p>
  By a result of <span class="name">Ibukiyama</span> (On Siegel modular varieties of level 3. Internat. J. Math. 2 (1991), 17-35, 
  <a href="http://www.ams.org/mathscinet-getitem?mr=MR1082834">MR1082834</a>),

 the ring <script type="math/tex">M_{*}(\Gamma_0(2))</script> of Siegel modular forms of degree 2  with respect to the group 
<a class="knowl-title" knowl="mf.siegel.group.gamma0">$\Gamma_0(2)$</a>
is generated by the following five generators, which involve the usage of 
<a class="knowl-title" knowl="mf.siegel.theta_constant">theta constants</a>.

<ul>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Gamma0_2', form = 'Generator_2', weight = 2, page = 'specimen') }}">$X$</a>, a form of weight 2,
with formula
$$X = ((\theta_{0000})^4+(\theta_{0001})^4+(\theta_{0010})^4+(\theta_{0011})^4)/4.$$
</li>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Gamma0_2', form = 'Generator_I_4', weight = 4, page = 'specimen') }}">$Y$</a>, a form  of weight 4,
with formula
$$Y = (\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011})^2.$$
</li>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Gamma0_2', form = 'Generator_II_4', weight = 4, page = 'specimen') }}">$Z$</a>, a form  of weight 4,
with formula
$$Z = ((\theta_{0100})^4-(\theta_{0110})^4)^2)/16384.$$
</li>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Gamma0_2', form = 'Generator_6', weight = 6, page = 'specimen') }}">$K$</a>, a form  of weight 6,
with formula
$$K = (\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111})^2/4096.$$
</li>
<li>
<a href="{{ url_for( 'ModularForm_GSp4_Q_top_level', group='Gamma0_2', form = 'Generator_19', weight = 19, page = 'specimen') }}">$\chi_{19}$</a>, a cusp form of weight 19,
with formula
$$\chi_{19} = \theta \theta'(8YZ-X^2T +YT +1024ZT+96T^2 - 8XK)/32,$$
where
<ul>
<li>
$\theta=\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011}\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111}$
</li>
<li>
$\theta' = ((\theta_{1000})^{12}+(\theta_{1001})^{12}+(\theta_{1100})^{12}+(\theta_{1111})^{12})/1536$
</li>
<li>
$T = (\theta_{0100}\theta_{0110})^{4}/256$
</li>
</ul>
</li>
</ul>

The generators $X, Y, Z, K$ are algebraically independent.
The ring of modular forms is

$$M(\Gamma_0(2)) =
\C[X,Y,Z,K] + \chi_{19}\,\C[X,Y,Z,K].$$


The ideal of cusp forms is ..??...

